Since if implies is symmetric, there exists an orthogonal matrix such that and is a diagonal matrix with the eigenvalues of along the diagonal and the columns of are the eigenvectors of . is called a unitary transformation. Since is orthogonal, and thus

Further, since when is nonsingular, is positive definite and thus has positive eigenvalues. Thus where has the square roots of the eigenvalues of on the diagonal.

Then where . So the problem is reduced to finding the eigenvalues and eigenvectors of a symmetric matrix There are efficient numerical methods for this.

Here is the issue when the identity matrix. It means that and thus is orthogonal. The problem is that any orthogonal matrix can serve as the eigenvector matrix for the identity matrix. Thus when the original is orthogonal, there is no way to identify it by looking at

This brings up a more general point. For any a matrix such that is not necessarily unique. The theory of unitary transformations says nothing about uniqueness.